predicting compressive stress on fibrous wool structures

I have searched for reference and handbook information (yes, google too)
describing the load behavior of randomly tangled wool-like fiber matte. The

interest is to design a wool-like material for elastic constant and
stiffness depending on the materials properties, volume fraction and
geometry (fiber thickness, cross section, length) of the fibers involved.
Materials will be inorganic fibers and not polymers or natural fibers.
Fibers will likely be curled or twisted and not straight. That should not
constrain the governing equations. I have no idea what governing equations
there may be. I expect there are empirical relations to use? There must be
some engineering work from ancient subjects like pillow making, steel wool,
filters or packaging. Fiber books don't cover this topic. No materials
books or chapters in them address such topics.
If there is a clue out there among you to resolve this, and you have read
this far, then let me add further refinements. Say I want to make a fibrous
matte much like Fiberfrax or the like. How might the mechanical
compressibility change if a heat treatment allows fiber interconnects to
form (sintering). Adhesives or sintering can ultimately make the matte into
a rigid structure. I wish to control stiffness or the spring constant but
maximize the compressed volume ratio. How would the properties change if I
substituted sapphire fibers for silica glass for example?
A book, web site, name of subject area or good keyword is all I ask.
Prof. K

The
equations
be
wool,
fibrous
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You have a fundamental problem, in that fibre mats themselves have no
intrinsic strength. They are all resilient and easily deformed. It is only
by mechanically bonding the fibres together that any web strength is
achieved. This may involve mechanical looping, as in needle mat, where
chopped strands are held in a veil support, or chemical bonding as typified
by emulsion or powder-bound chopped strand mat, or typical insulation fibre
blankets. Another approach is to make a paper-like material of dispersed
fibres.
You ask about the use of sapphire fibres. I presume that you have done a
Google search for "Saffil"?

You understand the problem. Thank You.
Saffil is like Nextel which is one of my materials. I should revisit Saffil
since the material might be engineered to a greater degree than their
commercial products. Thanks for activating that neuron that opened that
interconnect.
I wish to take advantage of the resiliency and predictably increase the
strength until the resiliency is reduced to my threshold. I had not heard
of powder-bound support, but it sounds like electrostatic binding. I wish
not to count on veil support. Instead, since I plan to avoid chopped
fibers, approach a similar state while including the coiling of fibers . I
seek 'loft' which is a term used in down (as in goose) which is another
material I wish to be biomimetic with.
Dr. K

Saffil
Powder binders are usually thermoplastic resins, used in chopped strand mat
and continuous strand mat to bind the fibre bundles together. Emulsion
binders are applied in aqueous emulsion, and have additional constituents to
modify the properties, like drape, softness, and so on. Usually PVAlcohol is
the base. In Insulation fibres, the binder used tended to be phenolic but I
don't know what they use these days.
Have you read J Gilbert Mohr's book, "Fiber Glass", Van Nostrand Reinhold,
ISBN 0-442-25447-4 ? My copy is dated 1978.

about microstructure and material properties, and how little we actually know
about it.
Your problem points it out quite nicely.
I came to this conclusion in studying the effects of porosity on
materials behavior.
First I concluded that we really didn't know what the heck porosity was.
I used open cell, closed cell and steel wool (actually plastic or metal
coil sink scrubber) materials to illustrate the point.
Secondly, I concluded that we often had no useful micromechanical or
microproperty theories...... and I used the same materials to illustrate
the point.
Dear Dr. K..... do you agree with this short summary?
Dr. Ashby hit the nail on the head in discussing mechanical applications as:
1) Tension (Or compression)
2) Bending
3) Torsion
In the systems of loose fibers such as steel wool, the mechanisms of
bending and perhaps even torsion dominate the micromechanical behavior.
This gives great softness or high compliance because of "lever" like
effects.
Eventually you can fill the voids enough that the compression response
of the matrix becomes dominant.
Or you can cross bond or cross link the filaments enough that eventually
the compression or tension modes of the fibers become dominant.
Theory sucks in understanding this.... and most other porosity effects,
except for the trivial.
Jim Buch

Yes, I fully concur.
I might add it seems fair to assume that all the stresses from fibers in
different orientations can be summed by superposition. The only problem is
quantitatively identifying the various fiber attributes of angle, bending,
bridging, torsion, cantilever etc. Then the various contributions can be
summed.
Certainly someone has done this years ago? Perhaps there are some empirical
ensemble equations?
How about twisting up some piano wire into a ball and measuring the spring
compression constant and then repeating with a double diameter wire of the
same volume? Then double the volume fraction. Will it be double the
stiffness? Who knows?
I am interested in curly long fibers and not straight chopped fibers of some
short length. The latter may be more tractable. I don't wish to address
the complication of particulate binders of varying strength and composition.
Should I continue an investigation in this; what journal would it best go
to? Now, if you have an answer for that, tell me what journal do I need to
search for related papers for the literature search? The answer to that is
what I need whether I publish or not.
Prof. K

The
involved.
not
equations
be
wool,
read
fibrous
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but
if I

jabbrer on about microstructure and material properties, and how little we
actually know about it.

I certainly don't know the answer to your question, but I have some
comments.
The Journal of the Textile Institute or Textile Research Journal might
have some relevant papers. See http://www.texi.org/pub.htm or
http://www.textileresearchjournal.com/ respectively.
I did some research on the fibres projecting from the surface of
textiles, many years ago. The stress-strain curve is roughly
exponential as one engages more and more projecting fibres, but I
stopped as soon as the body of the textile began to compress, which is
the point at which you wish to begin.
Keep friction in mind. A lot of what happens depends on friction at
the points of contact between fibres. In some circumstances, tacking
the fibres at the points of contact, by sintering or with another
material, makes very little difference to the compressive properties,
if friction was high to begin with.
The stiffness of the assembly is a very rapidly increasing function of
the packing density, because the average strut length decreases.
You might get some useful clues from "Particle Packing Characeristics"
by Randall M. German, ISBN 0-918404-83-5.
There was a paper in the Journal of the American Ceramic Society in
the last few years (sorry, I have lost the reference) that gave a
powder compression analogue of the general gas equation. There might
be some clues in that.
I think you have an intersting area to work in :)
Cheers
Alan Walker

From your message text I presume you want to design for compressive
elastic
modulus or compressive yield stress for a ceramic felt?
The people that make fiberglass insulation know a lot about how
fiber/fiber bonds affect elastic recovery from compression ("Loft
Recovery"). It is important that they be able to compress insulation
for shipping but have it
recover its fluffiness for installation.
Glues are usually applied during manufacture to lock fiber/fiber
junctions together thereby maintaining the microscopic geometric
arrangement of the felt's fibers. When such a felt is compressed, it
will return to its original
state if the bonds don't fail or the fibers don't permanently bend or
break.
It is no surprise that there is no ab-initio theory for predicting the
large deformation mechanical behavior of felts. Indeed there is not
even any reasonable way to describe the geometric details of felts.
You say that the fibers will probably be bent. How much are they bent?
What about corkscrew curves? These are very difficult geometries to
describe quantitatively. If we've no tools to describe the geometric
details of an
arrangement of fibers how can we hope to describe the mechanical
properties?
Fortunately there are a few things which are probably true about the
mechanical properties of felts:
If the fibers themselves are elastic (ie. recover their original state
when deforming stresses are removed) and the contact points between
fibers are either locked (ie. fibers cannot slide over each other) or
frictionless (ie. contact points have no mechanical consequence), then
the felt will be elastic because its geometry cannot be fundamentally
altered.
If any of the conditions in the preceeding paragraph are not met, the
felt will be inelastic.
Felts thus fall into two categories, elastic and non-elastic.
Sources of inelasticity:
Plastic deformation or fracture of fibers; these are obvious. Some
practical examples follow:
It is often found that heat treatment of ceramic fiber felts results
in drastic changes in elastic recovery because crystal growth can
severely weaken fibers. The weakened fibers break during deformation
resulting non-recoverable deformation (the felt's geometry is altered
during deformation).
I suppose it is possible that fiber/fiber sintering might increase
elasticity by locking fibers together; however, I fear that fiber
weakening by crystal growth might dominate.
If full elastic recovery is desired fiber/fiber junctions are usually
bonded with some kind of adhesive as in the case of fiberglass for
insulation.
Clearly, if stresses at bonded fiber/fiber junctions exceed a certain
level the bonds fail and fibers slide with respect to each other
resulting in a changed felt geometry and subsequent recovery.
A similar situation holds for fiber junctions locked by static
friction or electrostatic forces; above a certain stress fibers will
slide with respect to each other resulting in hysteresis effects. This
is the case for felts used for
piano string hammers.
Many people will attest to the effectiveness of hair conditioners in
easing combing out a rat's nest of hair. Similarly, surface chemistry
effects can drastically alter the mechanical properties of a felt. A
felted wool shirt feels different on a very humid day. These are
everyday examples of how a felt's mechanical properties can be altered
by surface chemistry effects.
The deformation and recovery behavior of a felt may depend strongly on
its past or present environment.
Elastic Felt descriptive parameters:
For small enough deformations, felts made from elastic fibers behave
elastically; ie. deformation is recoverable. It might be more precise
to say that an elastic response to deformation can be approached
asymptotically. I include this hedge because a fresh felt might not be
in its lowest energy configuration and small periodic mechanical
deformations may drive it towards an equilibrium configuration.
It is important to understand how felts scale dimensionally. That is,
given a particular felt structure, how do the felt properties change
if all dimensions are changed linearly? For example, what would happen
to a felt's elastic
modulus if all spacial dimensions are doubled?
Let's consider a simple case: a one dimensional array of fibers of
modulus Eo, like a bunch of parallel cables in tension. The cables are
x units apart and are of diameter d. A load F on a area of L*L is
shared between N=(L/x)*(L/x) cables. The stress on an individual
cable, So, is:
So=F/(N*(Pi*r*r)=F*(x*x)/((L*L)*Pi*r*r)
The strain is
e=So/Eo=(F/(L*L)/Eo)/(Pi*(r/x)*(r/x))
The nominal stress S is F/(L*L) and the nominal modulus is:
E=S/e=Eo*(Pi*(r/x)*(r/x))
The important scaling parameter in this simple case is the square of
the fiber radius to spacing ratio: (r/x)^2
It shows that the elastic properties of the cable assembly is
independent of magnification; the modulus stays the same if all
dimensions are changed equally.
Other important cases for fiber geometry include "S" shaped fibers
(see Therory of Elasticity, Timeshenko & Goodier, 3rd ed. art.33,
McGraw Hill, 1970) vertically loaded in the plane of the S.
In this case a new dimension can be introduced, R, the curvature of
the "S"; it turns out that the dimensional grouping that dictates the
effective modulus of a parallel assembly of squiggley cables x units
apart is proportional to:
Eo*(r/R)^2*(r/x)^2
Finally we look at fibers shaped like axially loaded helical springs
(see www.clc.tno.nl/projects/recent/spring.html for a nice description
of spring theory). In this case the spring's fiber radius is r, the
spacing between springs is x, the spring's loops are radius R, and the
loops are t units apart along the spring's axis (like inches per
turn); the appropriate dimensional grouping is:
Eo*(t/R)*(r/R)^2*(r/x)^2
We therefore expect that a felt's elastic modulus might be described
by a linear combination of terms involving the fiber modulus times a
nested hierarchy of dimensionless groups, (r/x), (r/R), and (t/R) that
describe the felt's geometry, like Eo*(r/x)^2*(a
+(r/R)^2*(b+(t/R)*(...))))
Note that in all cases the felt's effective modulus is proportional to
the fiber modulus, so holding structure constant and doubling fiber
modulus should double felt modulus. decreasing fiber spacing by a
factor of 2 quadruples modulus, etc...
Disclaimer: The above discussion pertaining to dimensional groups
describing the modulus of felts is entirely hypothetical and untested;
i.e. I made it up. I believe it to be reasonable and based on well
established mechanical principles. It assumes one dimensional loading
and does not consider the response of a spring loaded off-axis for
example.
I look forward to hearing critical commentary.

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